On non-linear optimization with a perturbed objective function

TL;DR

This paper develops a Lagrange multiplier theorem for optimization problems with imprecise objective functions using nonstandard analysis, providing algebraic tools to handle imprecisions directly.

Contribution

It introduces a novel approach to optimization with uncertain objectives by employing external numbers from nonstandard analysis, including imprecise differentiation and approximate theorems.

Findings

Derived a Lagrange multiplier theorem for imprecise objectives.

Utilized nonstandard analysis tools to handle imprecisions algebraically.

Provided a framework for optimization under uncertainty with rigorous mathematical foundations.

Abstract

A Lagrange multiplier theorem is derived for the case of an imprecise objective function and a precise constraint. The proof uses methods of analysis which deal in a direct, algebraic way with imprecisions. They include imprecise differentiation, and an approximate Fermat Lemma and Implicit Function Theorem. The tools are the external numbers of Nonstandard Analysis, which are models of Sorites imprecisions.